Calabi-Yau manifolds realizing symplectically rigid monodromy tuples

TL;DR

This paper introduces an iterative geometric construction of elliptically fibered Calabi-Yau manifolds that realizes all symplectically rigid Calabi-Yau differential operators with specific singularity structures.

Contribution

It provides a new method to construct Calabi-Yau families with prescribed monodromy and explicitly computes their Picard-Fuchs operators and periods.

Findings

Constructs families realizing all classified symplectically rigid operators.

Provides explicit formulas for periods and Picard-Fuchs operators.

Connects geometric constructions with differential operator classifications.

Abstract

We define an iterative construction that produces a family of elliptically fibered Calabi-Yau $n$-folds with section from a family of elliptic Calabi-Yau varieties of one dimension lower. Parallel to the geometric construction, we iteratively obtain for each family with a point of maximal unipotent monodromy, normalized to be at t=0, its Picard-Fuchs operator and a closed-form expression for the period holomorphic at t=0, through a generalization of the classical Euler transform for hypergeometric functions. In particular, our construction yields one-parameter families of elliptically fibered Calabi-Yau manifolds with section whose Picard-Fuchs operators realize all symplectically rigid Calabi-Yau differential operators with three regular singular points classified by Bogner and Reiter, but also non-rigid operators with four singular points.